Analysis and Pareto Frontier Based Tradeoff Design of an Integrated Magnetic Structure for a CLLC Resonant Converter

: This paper proposes an integrated magnetic structure for a CLLC resonant converter. With the proposed integrated magnetic structure, two resonant inductances and the transformer are integrated into one magnetic core, which improves the power density of the CLLC resonant converter. In the proposed integrated magnetic structure, two resonant inductances are decoupled with the transformer and can be adjusted by the number of turns in each inductance. Furthermore, two resonant inductances are coupled to reduce the number of turns in each inductance. As a result, the conduction loss can be reduced. The trade-off design of the integrated magnetic structure is carried out based on the Pareto optimization procedure. With the Pareto optimization procedure, both high efﬁciency and high-power density can be achieved. The proposed integrated magnetic structure is validated by theoretical analysis, simulations, and experiments.


Introduction
Bidirectional dc-dc converters are widely applied in the uninterrupted power supplies (UPS), electric vehicles, energy storage systems, smart grid power systems, fuel cells and supercapacitors hybrid systems [1][2][3][4] to achieve bidirectional power flows.
Of all the bidirectional dc-dc topologies, the CLLC resonant converter, which is shown in Figure 1, has become increasingly popular, due to its advantages of high efficiency, buck and boost capability, and bidirectional power transfer [5,6].In addition, compared with the dual active bridge (DAB) converter, where the Zero-Voltage-Switching is lost under light load conditions, the CLLC resonant converter can operate under Zero-Voltage-Switching (ZVS) of the primary switches and Zero-Current-Switching (ZCS) of the secondary switches over the whole load range [5,6].As a result, the switching loss can be greatly reduced.Due to the soft switching characteristics, the CLLC resonant converter can operate at a high frequency, which results in a high-power density.Furthermore, the voltage stress of the power switches in the CLLC resonant converter is confined to the input voltage and output voltage without any clamp circuit [5].Therefore, the CLLC resonant converters have been investigated and reported extensively.
In the CLLC topology level, the working principles, equations of main parameters, synchronous rectifications, and the controls of CLLC resonant converters have been investigated [5,[7][8][9].The operation principles and the equations of the main parameters are well stated in [5,7,8].To reduce the conduction loss, a novel synchronous rectification scheme based on small phase shift technique is proposed in [9].In the CLLC topology level, the working principles, equations of main parameters, synchronous rectifications, and the controls of CLLC resonant converters have been investigated [5,[7][8][9].The operation principles and the equations of the main parameters are well stated in [5,7,8].To reduce the conduction loss, a novel synchronous rectification scheme based on small phase shift technique is proposed in [9].
In the component level, in order to improve the power density, the wide bandgap semiconductor devices, such as the Gallium Nitride (GaN) and Silicon Carbide (SiC) switches, have been utilized to reduce the power loss of the switches and increase the switching speed.However, with the reduction of the power loss of the switches, the magnetic design is still very challenging mainly due to the number of the magnetic cores in the CLLC resonant converter, where two resonant inductances and a transformer are required [10][11][12].
The most basic way to build two resonant inductances and a transformer for a CLLC resonant converter is to use three detached magnetic cores, with two magnetic cores for the resonant inductances and another for the transformer, which is shown in Figure 2a [5,9].The advantage of the basic way is that the two resonant inductances can be adjusted independently based on the optimal design of the CLLC resonant converter.However, the application of the detached magnetic cores will lead to the large volume of the converter, which will reduce the power density.The conventional integrated magnetic structure as shown in Figure 2b, where the leakage inductances of the transformer are used as the resonant inductances, is applied in the CLLC resonant converter to increase the power density [12][13][14].The leakage energy is stored between the primary winding and the secondary winding.The leakage inductance (Lr) can be adjusted by the distance of the primary winding and secondary winding (dw), In the component level, in order to improve the power density, the wide bandgap semiconductor devices, such as the Gallium Nitride (GaN) and Silicon Carbide (SiC) switches, have been utilized to reduce the power loss of the switches and increase the switching speed.However, with the reduction of the power loss of the switches, the magnetic design is still very challenging mainly due to the number of the magnetic cores in the CLLC resonant converter, where two resonant inductances and a transformer are required [10][11][12].
The most basic way to build two resonant inductances and a transformer for a CLLC resonant converter is to use three detached magnetic cores, with two magnetic cores for the resonant inductances and another for the transformer, which is shown in Figure 2a [5,9].The advantage of the basic way is that the two resonant inductances can be adjusted independently based on the optimal design of the CLLC resonant converter.However, the application of the detached magnetic cores will lead to the large volume of the converter, which will reduce the power density.
v CD (t) In the CLLC topology level, the working principles, equations of main parameters, synchronous rectifications, and the controls of CLLC resonant converters have been investigated [5,[7][8][9].The operation principles and the equations of the main parameters are well stated in [5,7,8].To reduce the conduction loss, a novel synchronous rectification scheme based on small phase shift technique is proposed in [9].
In the component level, in order to improve the power density, the wide bandgap semiconductor devices, such as the Gallium Nitride (GaN) and Silicon Carbide (SiC) switches, have been utilized to reduce the power loss of the switches and increase the switching speed.However, with the reduction of the power loss of the switches, the magnetic design is still very challenging mainly due to the number of the magnetic cores in the CLLC resonant converter, where two resonant inductances and a transformer are required [10][11][12].
The most basic way to build two resonant inductances and a transformer for a CLLC resonant converter is to use three detached magnetic cores, with two magnetic cores for the resonant inductances and another for the transformer, which is shown in Figure 2a [5,9].The advantage of the basic way is that the two resonant inductances can be adjusted independently based on the optimal design of the CLLC resonant converter.However, the application of the detached magnetic cores will lead to the large volume of the converter, which will reduce the power density.The conventional integrated magnetic structure as shown in Figure 2b, where the leakage inductances of the transformer are used as the resonant inductances, is applied in the CLLC resonant converter to increase the power density [12][13][14].The leakage energy is stored between the primary winding and the secondary winding.The leakage inductance (Lr) can be adjusted by the distance of the primary winding and secondary winding (dw), The conventional integrated magnetic structure as shown in Figure 2b, where the leakage inductances of the transformer are used as the resonant inductances, is applied in the CLLC resonant converter to increase the power density [12][13][14].The leakage energy is stored between the primary winding and the secondary winding.The leakage inductance (L r ) can be adjusted by the distance of the primary winding and secondary winding (d w ), while the magnetizing inductance (L m ) is adjusted by the thickness of the air gap (d a ).With this magnetic integration method, only one magnetic core is required.However, the shortcomings of this magnetic structure are summarized as follows: Firstly, the extra distance (d w ) is required to achieve the desired resonant inductances, which will increase the volume of the magnetic core and the corresponding core loss.Secondly, due to the high leakage flux between primary and secondary winding, the conductors will suffer more severe proximity effect, which will increase the copper loss.
Finally, it should be noted that the range of the leakage inductances is limited by the height of the magnetic core (h c ), which means the desired resonant inductances may not be achieved when large resonant inductances are required.
In order to solve the above problems, an integrated magnetic structure is proposed for the CLLC resonant converter, which is illustrated in Figure 2c.In the proposed integrated magnetic structure, instead of using the leakage inductances as the resonant inductances, two extra inductances, L r1 and L r2 are introduced to the magnetic core, which are coupled with each other but decoupled with L m .The sums of the self-inductance (L r1 , L r2 ) and the mutual inductance (M 12 ) are used as the resonant inductances (L rp , L rs ).In addition, L rp and L rs can be adjusted by the number of turns (N r1 and N r2 ) and the air gap in the side legs (d s ) while L m can be adjusted by the number of turns (N p ) and the air gap in the central leg (d c ).The advantages of the proposed integrated magnetic structure are summarized as follows: Firstly, only one magnetic core is required, which is the same with the conventional integrated magnetic structure shown in Figure 2b.
Secondly, compared with Figure 2b, d w is not required anymore and the volume of the magnetic core will be reduced.The proximity effect in Figure 2b will be reduced as well.
Thirdly, the resonant inductances can be easily increased by adding more turns for L r1 and L r2 , so the range of the resonant inductances is expanded.As a result, the optimal design of the CLLC resonant converter can be achieved although large resonant inductances are required.
Finally, L r1 and L r2 are positively coupled to enhance the resonant inductances.As a result, the number of turns (N r1 and N r2 ) is reduced compared with using uncoupled inductances.As a result, the copper loss is reduced as well.
However, it is important to point out that in the proposed integrated magnetic structure, due to the positive coupling between L r1 and L r2 , the magnetic flux density in the side legs will increase, which will result in more core loss.By increasing the cross-sectional area, both the magnetic flux density and the core loss will decrease.However, the core volume will increase.As a result, the core loss and the core volume always contradict each other.In order to further optimize the proposed integrated magnetic structure, with full consideration of the width of the central leg, w c , the width of the side legs, w s , and the depth of the magnetic core, w d , the tradeoff design between the core volume and core loss is also investigated, which is based on the multi-objective optimization method, the Pareto optimization procedure.With the Pareto optimization procedure, the optimal design of the proposed integrated magnetic structure can be easily carried out for a specific application.
The rest of this paper is arranged as follows.The conventional integrated magnetic structure for the CLLC resonant converter is reviewed in Section 2. In Section 3, the operation principles and the analysis of the proposed integrated magnetic structure are elaborated and the equations of L rp , L rs and L m are derived.In Section 4, the Pareto Frontier based multiobjective optimization of the proposed integrated magnetic structure is also investigated.In Section 5, the proposed magnetic structure and the multi-objective optimization are validated by the simulations and experiments.Section 6 contains a brief conclusion.

Review of the Conventional Method of Magnetic Integration in the CLLC Resonant Converter: Using the Leakage Inductances as the Resonant Inductances
The conventional integrated magnetic structure in a CLLC resonant converter is to use the leakage inductances as the resonant inductances, as shown in Figure 2b [12][13][14].In this section, this conventional method is reviewed.The general equation of the leakage inductance of this method is derived and the shortcomings of this method are analyzed.
As shown in Figure 3, in order to increase the power density by reducing the number of the magnetic components, the leakage inductances of the planar transformer are utilized as the resonant inductances.The desired leakage inductances (L rp and L rs ) are achieved by changing the distance between the primary winding and the secondary winding (d w ).
In this section, this conventional method is reviewed.The general equation of the leakage inductance of this method is derived and the shortcomings of this method are analyzed.
As shown in Figure 3, in order to increase the power density by reducing the number of the magnetic components, the leakage inductances of the planar transformer are utilized as the resonant inductances.The desired leakage inductances (Lrp and Lrs) are achieved by changing the distance between the primary winding and the secondary winding (dw).The winding configuration of a planar transformer and its corresponding magnetomotive force (MMF) are shown in Figure 3.The primary winding has np turns, which are distributed in mp layers while the secondary winding has ns turns, which are distributed in ms layers.The distance between the primary winding and the secondary winding is dw, which is adjusted to achieve the optimal leakage inductances.The air gap (da) is added to achieve the desired magnetizing inductance (Lm).
Assuming the current excitation to the primary winding is ip, based on the distribution of MMF in Figure 3, the magnetic energy stored in the window can be calculated by [15][16][17]: where Wd is the depth of the window, Ww is the width of the window, dp and ds are the thickness of the primary winding and secondary winding, respectively, and di is the thickness of the insulation.
According to the relation between leakage inductance and magnetic energy, leakage inductances can be calculated by:  The winding configuration of a planar transformer and its corresponding magnetomotive force (MMF) are shown in Figure 3.The primary winding has n p turns, which are distributed in m p layers while the secondary winding has n s turns, which are distributed in m s layers.The distance between the primary winding and the secondary winding is d w , which is adjusted to achieve the optimal leakage inductances.The air gap (d a ) is added to achieve the desired magnetizing inductance (L m ).
Assuming the current excitation to the primary winding is i p , based on the distribution of MMF in Figure 3, the magnetic energy stored in the window can be calculated by [15][16][17]: where W d is the depth of the window, W w is the width of the window, d p and d s are the thickness of the primary winding and secondary winding, respectively, and d i is the thickness of the insulation.
According to the relation between leakage inductance and magnetic energy, leakage inductances can be calculated by: It can be concluded from Equation (2) that the leakage inductances can be changed by d w , which can be calculated by: However, it is important to note the shortcomings of this method, which are summarized as follows: Firstly, with increased d w , the volume of the magnetic core will increase as well, which will not only reduce the power density of the converter, but also cause extra core loss.
Secondly, in order to achieve the desired leakage inductances, plenty of leakage fluxes should be stored between the primary winding and the secondary winding.However, the stored leakage flux will aggravate the proximity effect, which will cause more copper loss.
Finally, it should be noted that d w is limited by the height of the window.As a result, the range of the leakage inductances is also limited by the window.

Proposed Integrated Magnetic Structure for the CLLC Resonant Converters
In this section, an integrated magnetic structure is proposed for a CLLC resonant converter, where two resonant inductances and a transformer are integrated in a magnetic core.At first, the working principles and the equations are elaborated.After that, the proposed Energies 2021, 14, 1756 5 of 19 integrated magnetic structure is compared with the conventional integrated magnetic structure to further show the advantages of the proposed integrated magnetic structure.

Proposed Integrated Magnetic Structure for the CLLC Resonant Converters
The proposed integrated magnetic structure and the corresponding magnetic flux are shown in Figure 4, where the primary winding and the secondary winding are wounded on the center leg while two extra inductances (L r1 and L r2 ) are wounded on the side legs.It is worthy to point out that the sums of the self-inductance (L r1 , L r2 ) and the mutual inductance (M 12 ) are used as the resonant inductances (L rp , L rs ).
Finally, it should be noted that dw is limited by the height of the window.As a result, the range of the leakage inductances is also limited by the window.

Proposed Integrated Magnetic Structure for the CLLC Resonant Converters
In this section, an integrated magnetic structure is proposed for a CLLC resonant converter, where two resonant inductances and a transformer are integrated in a magnetic core.At first, the working principles and the equations are elaborated.After that, the proposed integrated magnetic structure is compared with the conventional integrated magnetic structure to further show the advantages of the proposed integrated magnetic structure.

Proposed Integrated Magnetic Structure for the CLLC Resonant Converters
The proposed integrated magnetic structure and the corresponding magnetic flux are shown in Figure 4, where the primary winding and the secondary winding are wounded on the center leg while two extra inductances (Lr1 and Lr2) are wounded on the side legs.It is worthy to point out that the sums of the self-inductance (Lr1, Lr2) and the mutual inductance (M12) are used as the resonant inductances (Lrp, Lrs).

Primary
Secondary The primary and secondary winding, which are wounded on the central leg, have Np and Ns turns, respectively.The magnetic flux generated by Lm, ϕm, is shown by the blue dash line.
The inductance, Lr1, has Nr1 turns, with half of the turns are on the left leg while the other half are on the right leg.The way how the two halves are connected is shown in Figure 4.The flux generated by Lr1, ϕr1, is shown by the red dash line.
The inductance, Lr2, has Nr2 turns, with half of the turns are on the left leg while the other half are on the right leg.The way how the two halves are connected is shown in The primary and secondary winding, which are wounded on the central leg, have N p and N s turns, respectively.The magnetic flux generated by L m , φ m , is shown by the blue dash line.
The inductance, L r1 , has N r1 turns, with half of the turns are on the left leg while the other half are on the right leg.The way how the two halves are connected is shown in Figure 4.The flux generated by L r1 , φ r1 , is shown by the red dash line.
The inductance, L r2 , has N r2 turns, with half of the turns are on the left leg while the other half are on the right leg.The way how the two halves are connected is shown in Figure 4.The flux generated by L r2 , φ r2 , is shown by the green dash line.The equivalent magnetic circuit with the excitation is shown in Figure 5.
Assuming the current excitation of Lr1 is ir1, the equivalent magnetic circuit is shown in Figure 5a.
Based on magnetic circuit analysis, the following equations can be derived: Neglecting the leakage flux, L m , L r1 , L r2 , the mutual inductance between L r1 and L r2 , M 12 , the mutual inductance between L r1 and L m , M 1m and the mutual inductance between L r2 and L m , M 2m , will be derived as follows.
Assuming the current excitation of L r1 is i r1 , the equivalent magnetic circuit is shown in Figure 5a.Based on magnetic circuit analysis, the following equations can be derived: where φ s1 , φ s2 are the magnetic flux in the left-side leg and right-side leg, respectively.R c and R s are the magnetic reluctances of the central leg, side legs and yokes (including the air gaps).
According to Equation ( 4), φ s1 and φ s2 are: According to the definitions of self-inductance and mutual inductance, with (5), L r1 , M 1m and M 12 can be calculated by: As a result, L m and L r1 are decoupled.Similarly, the current excitation of L r2 is i r2 and the equivalent magnetic circuit is shown in Figure 5b.Based on the magnetic circuit analysis, the following equations can be derived: According to the definitions of self-inductance and mutual inductance, with (7), L r2 , M 2m and M 21 can be calculated by: As a result, L m and L r2 are decoupled.
The current excitation of L m is i m and the equivalent magnetic circuit is shown in Figure 5c.L m can be calculated by: Due to the coupling between L r1 and L r2 , the actual resonant inductances, L rp and L rs are: In this part, the proposed integrated magnetic structure is compared with the conventional integrated magnetic structure, where the leakage inductances are used as the resonant inductances.The parameters of the CLLC resonant converter are listed in Table 1.The magnetic core selected for the CLLC resonant converter is ELP 64/10/50 with N87 from TDK.Two magnetic cores are used in parallel to increase the cross-sectional area.The parameters of the magnetic core are shown in Table 1 as well.
In order to make a fair comparison, the two integrated magnetic structures share the same winding configuration, where primary winding is distributed into 4 layers, with each layer consisting of 4 turns.In order to achieve the desired L m , the thickness of the air gap, d a , is set as 0.14 mm.In the conventional method, in order to achieve the desired leakage inductances, the distance between the primary winding and the secondary winding, d w , is calculated by Equation ( 3), which is: As a result, in order to provide sufficient space for the distance between the primary winding and the secondary winding, a combination of ELP 64/10/50 with ELP 64/10/50 must be adopted, which is shown in Figure 6a.As for the proposed integrated magnetic structure, considering the coupling between Lr1 and Lr2, the number of turns of Lr1 and Lr2 are: Due to the lower height of this magnetic structure, a combination of ELP 64/10/50 with I 64/5/50 can be adopted, which is shown in Figure 6b.
It can be seen from Figure 6 that compared with Figure 6a, the volume of the magnetic core in Figure 6b is reduced from 41.500 mm 3 to 36.200 mm 3 and the height is reduced from 20.2 to 15.3 mm.In addition, it can be seen from Figure 6a that the height of the window has been totally utilized to achieve the desired resonant inductances, which means with the conventional method, the range of the resonant inductances is limited by the height of the window.However, with the proposed integrated magnetic structure, higher resonant inductances can still be easily achieved.
However, it is worth to point out that the flux density in the proposed integrated magnetic structure (Figure 6b) is higher than that in the conventional integrated magnetic structure (Figure 6a).This problem can be solved by optimal design, which will be elaborated in the following section.

Pareto Frontier Based Tradeoff Design of the Proposed Integrated Magnetic Structure
As mentioned above, the core volume and the core loss always contradict each other.In this part, based on the Pareto Frontier, the tradeoff design of the proposed integrated magnetic structure between the core volume and the core loss is investigated.Since the dimensions of the window are determined by the winding structure, during the tradeoff design process, the width (Ww) and the height (Wh) of the window are viewed as constant.As for the proposed integrated magnetic structure, considering the coupling between L r1 and L r2 , the number of turns of L r1 and L r2 are: Due to the lower height of this magnetic structure, a combination of ELP 64/10/50 with I 64/5/50 can be adopted, which is shown in Figure 6b.
It can be seen from Figure 6 that compared with Figure 6a, the volume of the magnetic core in Figure 6b is reduced from 41.500 mm 3 to 36.200 mm 3 and the height is reduced from 20.2 to 15.3 mm.In addition, it can be seen from Figure 6a that the height of the window has been totally utilized to achieve the desired resonant inductances, which means with the conventional method, the range of the resonant inductances is limited by the height of the window.However, with the proposed integrated magnetic structure, higher resonant inductances can still be easily achieved.
However, it is worth to point out that the flux density in the proposed integrated magnetic structure (Figure 6b) is higher than that in the conventional integrated magnetic structure (Figure 6a).This problem can be solved by optimal design, which will be elaborated in the following section.

Pareto Frontier Based Tradeoff Design of the Proposed Integrated Magnetic Structure
As mentioned above, the core volume and the core loss always contradict each other.In this part, based on the Pareto Frontier, the tradeoff design of the proposed integrated magnetic structure between the core volume and the core loss is investigated.Since the dimensions of the window are determined by the winding structure, during the tradeoff Energies 2021, 14, 1756 8 of 19 design process, the width (W w ) and the height (W h ) of the window are viewed as constant.As shown in Figure 7, the tradeoff design focuses three parameters: the width of the side legs, w s , the width of the center leg, w c , and the depth of the window, w d .In the first part, the total magnetic core loss of the proposed integrated magnetic structure is derived, which will be used for the following tradeoff design.In the second part, the effects of w s , w c , and w d on the total core loss are investigated and the corresponding simulations are carried out to validate the analysis.In the third part, the total core loss is optimized by Pareto Frontier with fully considering w s , w c , and w d .
Energies 2021, 14, x FOR PEER REVIEW 9 of 20  where Ir1pk, Ir2pk and Impk are the peak values of ir1, ir2 and im, respectively.VL, VC and VR are illustrated by Figure 7. Aes and Aec are the cross-sectional area of the side leg and central leg, which are also shown in Figure 7.
Based on the Steinmetz's formula, combining Equation ( 13), the total core loss of the magnetic core can be calculated by: where kc, α and β are the parameters of the magnetic core, which can be found from the datasheet.The equation of total core loss Pc will be used for the further optimal design.

Effects of ws, wc, and wd on the Total Core Loss Pc
The effects of ws, wc, and wd on Pc are studied in this part.The selected magnetic core is ELP 64/10/50 with N87, from TDK.In order to increase the power rating, two magnetic cores are used in parallel to increase the cross-sectional area.The sizes of the magnetic core are shown in Table 2, which will be used as the base values in the following comparison.With i r1 , i r2 and i m applied to L r1 , L r2 and L m , the flux density of V L , V R and V C , B Lpk , B Rpk and B Cpk , can be calculated by: where I r1pk , I r2pk and I mpk are the peak values of i r1 , i r2 and i m , respectively.V L , V C and V R are illustrated by Figure 7.A es and A ec are the cross-sectional area of the side leg and central leg, which are also shown in Figure 7.
Based on the Steinmetz's formula, combining Equation ( 13), the total core loss of the magnetic core can be calculated by: where k c , α and β are the parameters of the magnetic core, which can be found from the datasheet.The equation of total core loss P c will be used for the further optimal design.

Effects of w s , w c , and w d on the Total Core Loss P c
The effects of w s , w c , and w d on P c are studied in this part.The selected magnetic core is ELP 64/10/50 with N87, from TDK.In order to increase the power rating, two magnetic cores are used in parallel to increase the cross-sectional area.The sizes of the magnetic core are shown in Table 2, which will be used as the base values in the following comparison.In order to make a fair comparison, all the cases in the following share the same winding configuration, which is the same with that in Figure 6b.
The current excitations are from the circuit simulation of a CLLC resonant converter.The parameters of the CLLC resonant converter as well as the current excitations are shown in Table 3.The current excitations are from the circuit simulation of a CLLC resonant converter.The parameters of the CLLC resonant converter as well as the current excitations are shown in Table 3.It can be seen from Figure 8 that although wc is reduced as shown in Figure 8a or increased as shown in Figure 8c the magnetic flux density is still low, which may be further optimized.This is because in the proposed integrated magnetic structure, the flux mainly concentrates on the side legs.The variation of Pc with different wc is calculated, which is shown in Figure 8d.All the parameters in Figure 8d are in per-unit-value.The original wc, wc0 (10.4 mm) and the corresponding core loss, Pc0 (5.8 W) are used as the based value.
It can be seen from Figure 8d considering the volume and the core loss, the central leg of the magnetic core can be further optimized.
(b) Effect of ws on Core Loss It can be seen from Figure 8 that although w c is reduced as shown in Figure 8a or increased as shown in Figure 8c the magnetic flux density is still low, which may be further optimized.This is because in the proposed integrated magnetic structure, the flux mainly concentrates on the side legs.The variation of P c with different w c is calculated, which is shown in Figure 8d.All the parameters in Figure 8d are in per-unit-value.The original w c , w c0 (10.4 mm) and the corresponding core loss, P c0 (5.8 W) are used as the based value.
It can be seen from Figure 8d considering the volume and the core loss, the central leg of the magnetic core can be further optimized.
(b) Effect of w s on Core Loss In the second case, the width of the side legs, w s , is changed from 20 to 140% of the base value (5.2 mm), while other parameters are kept the same with the base values.Simulations are carried out in Ansys Maxwell and the distributions of the magnetic flux density are shown in Figure 9.In the second case, the width of the side legs, ws, is changed from 20 to 140% of the base value (5.2 mm), while other parameters are kept the same with the base values.Simulations are carried out in Ansys Maxwell and the distributions of the magnetic flux density are shown in Figure 9.It can be seen from Figure 9 that compared with Figure 9b as ws is reduced in Figure 9a the magnetic flux density in the side leg will increase.In contrast, compared with Figure 9b, in Figure 9c, as ws is increased, the magnetic flux density in the side leg will decrease.The variation of Pc with different ws is calculated, which is shown in Figure 9d.All the parameters in Figure 9 are in per-unit-value.The original ws, ws0 (5.2 mm) and the corresponding core loss, Pc0 (5.8 W) are used as the base values.
It can be seen from Figure 9d that with larger ws, the magnetic flux density in the side leg will decrease and the core loss will decrease.It is because that the flux in the side legs is enhanced due to coupling between Lr1 and Lr2.As a result, in order to achieve better performance for a specific application, the side legs need further optimization.
(c) Effect of wd on Core Loss In the third case, the depth of the magnetic core, wd, is changed from 20 to 140% of the base value (101.6 mm), while other parameters are kept the same with the base values.Simulations are carried out in Ansys Maxwell and the distributions of the magnetic flux density are shown in Figure 10.It can be seen from Figure 9 that compared with Figure 9b as w s is reduced in Figure 9a the magnetic flux density in the side leg will increase.In contrast, compared with Figure 9b, in Figure 9c, as w s is increased, the magnetic flux density in the side leg will decrease.The variation of P c with different w s is calculated, which is shown in Figure 9d.All the parameters in Figure 9 are in per-unit-value.The original w s , w s0 (5.2 mm) and the corresponding core loss, P c0 (5.8 W) are used as the base values.
It can be seen from Figure 9d that with larger w s , the magnetic flux density in the side leg will decrease and the core loss will decrease.It is because that the flux in the side legs is enhanced due to coupling between L r1 and L r2 .As a result, in order to achieve better performance for a specific application, the side legs need further optimization.
(c) Effect of w d on Core Loss In the third case, the depth of the magnetic core, w d , is changed from 20 to 140% of the base value (101.6 mm), while other parameters are kept the same with the base values.Simulations are carried out in Ansys Maxwell and the distributions of the magnetic flux density are shown in Figure 10.
It can be seen from Figure 10 that compared with Figure 10b, as w d is reduced in Figure 10a, the magnetic flux density in the side leg will increase.In contrast, compared with Figure 10b, as w d is increased in Figure 10c, the magnetic flux density in the center leg will decrease.The variation of P c with different w d is calculated, which is shown in Figure 10.
It can be seen from Figure 10 that with the increase of w d , the core loss will decrease.It is because that the flux in the side legs is enhanced due to coupling between L r1 and L r2 .As a result, in order to achieve better performance for a specific application, the depth of the core need further optimization, which is similar to the effect of w s .
is enhanced due to coupling between Lr1 and Lr2.As a result, in order to achieve better performance for a specific application, the side legs need further optimization.
(c) Effect of wd on Core Loss In the third case, the depth of the magnetic core, wd, is changed from 20 to 140% of the base value (101.6 mm), while other parameters are kept the same with the base values.Simulations are carried out in Ansys Maxwell and the distributions of the magnetic flux density are shown in Figure 10.It can be seen from Figure 10 that compared with Figure 10b, as wd is reduced in Figure 10a, the magnetic flux density in the side leg will increase.In contrast, compared with Figure 10b, as wd is increased in Figure 10c, the magnetic flux density in the center leg will decrease.The variation of Pc with different wd is calculated, which is shown in Figure 10.
It can be seen from Figure 10 that with the increase of wd, the core loss will decrease.It is because that the flux in the side legs is enhanced due to coupling between Lr1 and Lr2.As a result, in order to achieve better performance for a specific application, the depth of the core need further optimization, which is similar to the effect of ws.
Based on the above analysis, it can be concluded that the core loss of the magnetic core is affected by ws, wc, and wd.In order to achieve low core loss, ws, wc, and wd need further optimization.However, it is important to point out that the core loss and the volume of the magnetic core always contradict each other: lower core loss means larger volume of the magnetic core while lower volume of the magnetic core means higher core loss.Considering both core loss and the volume of the core, in the following part, the tradeoff design of the proposed integrated magnetic structure based on Pareto Frontier will be investigated.

Tradeoff Design of the Proposed Integrated Magnetic Structure Based on Pareto Optimization Procedure
The Pareto optimization [18,19] is utilized to identify the best performing proposed integrated magnetic structure for a CLLC resonant converter.The specifications of the converter and the magnetic material are required for the calculations of the volume and power loss of the magnetic core.Each parameter within the limits of the design space is swept individually, resulting in dataset where each point represents an integrated magnetic design and its corresponding performance, including the volume and power loss of the core.By analyzing the volume and core loss, the Pareto frontier can be revealed, which represents the lowest achievable magnetic core loss for a given volume.
(a) Flowchart of Pareto optimization procedure for the proposed integrated magnetic structure The flowchart of the proposed integrated magnetic structure Pareto optimization procedure is shown in Figure 11.Based on the above analysis, it can be concluded that the core loss of the magnetic core is affected by w s , w c , and w d .In order to achieve low core loss, w s , w c , and w d need further optimization.However, it is important to point out that the core loss and the volume of the magnetic core always contradict each other: lower core loss means larger volume of the magnetic core while lower volume of the magnetic core means higher core loss.Considering both core loss and the volume of the core, in the following part, the tradeoff design of the proposed integrated magnetic structure based on Pareto Frontier will be investigated.

Tradeoff Design of the Proposed Integrated Magnetic Structure Based on Pareto Optimization Procedure
The Pareto optimization [18,19] is utilized to identify the best performing proposed integrated magnetic structure for a CLLC resonant converter.The specifications of the converter and the magnetic material are required for the calculations of the volume and power loss of the magnetic core.Each parameter within the limits of the design space is swept individually, resulting in dataset where each point represents an integrated magnetic design and its corresponding performance, including the volume and power loss of the core.By analyzing the volume and core loss, the Pareto frontier can be revealed, which represents the lowest achievable magnetic core loss for a given volume.
(a) Flowchart of Pareto optimization procedure for the proposed integrated magnetic structure The flowchart of the proposed integrated magnetic structure Pareto optimization procedure is shown in Figure 11.
The details of each step are as follows: Step 1: Initialization of Pareto optimization procedure The operation conditions of the CLLC resonant converter, O = {I r1pk , I r2pk , I mpk }, which are derived from the circuit simulations, are set.The design space, W d = {40.64mm ≤ w d ≤ 160.4 mm}, W c = {4.08 mm ≤ w c ≤ 15.3 mm}, W s = {2.08 mm ≤ w s ≤ 7.80 mm}, which are used to limit the dimensions of the magnetic core, are defined.
Step 2: Calculations of the dominated number of each individual (n p ) and crowding distance (C d ) In this step, at first, the multi-objective values, the volume of the magnetic core, V c , and the core loss, P c , are calculated.n p and C d of each individual are calculated with V c and P c based on non-dominated sorting.
After generation of X newj , X j and X newj are combined together to form a population X 2Nj , which includes 2N individuals.
Step 4: Recycling of Step 2 and Step 3 In this step, if the maximum iteration number is reached, the recycling process stops.Otherwise, N individuals will be chosen for X j as the next generation and the new recycling process of Step 2 and Step 3 will begin.
Step 5: Generate the Pareto frontier After the maximum iteration number is reached, the recycling process stops, and the Pareto frontier will be generated.In this part, the Pareto optimization is used for the design of the proposed integrated magnetic structure in a CLLC resonant converter.The specifications of the CLLC resonant converter are shown in Table 3.The magnetic material is N87, from TDK.With the flowchart shown in Figure 11, the generated Pareto Frontier is shown in Figure 12.The conventional design, where the leakage inductances are used as the resonant inductances is also shown in Figure 12 (Point B) for comparison.
(b) Application of Pareto optimization procedure for the proposed integrated magnetic structure In this part, the Pareto optimization is used for the design of the proposed integrated magnetic structure in a CLLC resonant converter.The specifications of the CLLC resonant converter are shown in Table 3.The magnetic material is N87, from TDK.With the flowchart shown in Figure 11, the generated Pareto Frontier is shown in Figure 12.The conventional design, where the leakage inductances are used as the resonant inductances is also shown in Figure 12  It can be seen from Figure 12 that with the increase of the magnetic core volume, the core loss will decrease.However, the volume of the magnetic core cannot be increased unboundedly.In a specific application, there is trade-off design between the volume and the power loss.
In addition, under the same volume condition, the core loss on the Pareto frontier is lower than the conventional design, which means that with optimal design, the proposed integrated magnetic structure can achieve not only high-power density but also low power loss.The optimal dimensions of the magnetic core are ws = 3.82 mm, wc = 6.64 mm, and wd = 152.39mm.
The conventional design and the optimal design based on Pareto Frontier will be validate by simulations in the following section.

Simulation and Experimental Validations of the Proposed Integrated Magnetic Structure and the Pareto Optimization Procedure
In this section, the proposed integrated magnetic structure and the Pareto optimization procedure are validated by the hybrid electromagnetic simulations and experiments.Three cases are considered in this section:


Conventional design, Point B, as shown in Figure 12, where the leakage inductances are used as the resonant inductances.


Unoptimized design of the proposed integrated magnetic structure, Point C, as shown in Figure 12.  Optimal design of the proposed integrated magnetic structure based on Pareto frontier, Point A, as shown in Figure 12.It can be seen from Figure 12 that with the increase of the magnetic core volume, the core loss will decrease.However, the volume of the magnetic core cannot be increased unboundedly.In a specific application, there is trade-off design between the volume and the power loss.
In addition, under the same volume condition, the core loss on the Pareto frontier is lower than the conventional design, which means that with optimal design, the proposed integrated magnetic structure can achieve not only high-power density but also low power loss.The optimal dimensions of the magnetic core are w s = 3.82 mm, w c = 6.64 mm, and w d = 152.39mm.
The conventional design and the optimal design based on Pareto Frontier will be validate by simulations in the following section.

Simulation and Experimental Validations of the Proposed Integrated Magnetic Structure and the Pareto Optimization Procedure
In this section, the proposed integrated magnetic structure and the Pareto optimization procedure are validated by the hybrid electromagnetic simulations and experiments.Three cases are considered in this section:

•
Conventional design, Point B, as shown in Figure 12, where the leakage inductances are used as the resonant inductances.

•
Unoptimized design of the proposed integrated magnetic structure, Point C, as shown in Figure 12.

•
Optimal design of the proposed integrated magnetic structure based on Pareto frontier, Point A, as shown in Figure 12.

Simulation Validations
The parameters of the CLLC resonant converter are shown in Table 3.The simulated magnetic flux density of the conventional design, unoptimized design of the proposed integrated magnetic structure, and optimized design of the proposed integrated magnetic structure are shown in Figure 13a-c, respectively.
Based on the distributions of the magnetic flux density, the core loss of each design is calculated, which is shown in Figure 13d.
It can be seen from Figure 13d that the core loss of the unoptimized design of the proposed integrated magnetic structure is higher than the conventional design.This is because the current of L rp and L rs flowing through the side legs of the magnetic core causes extra core loss.However, with the Pareto optimization procedure, the core loss of the optimized design of the propose integrated magnetic structure is reduced, which is lower than the conventional design.
As a result, it can be concluded that with the Pareto optimization procedure, the proposed integrated magnetic structure can enjoy higher power density and lower core loss compared with the conventional design.In the following part, the conventional design, un-optimized proposed integrated magnetic structure and the Pareto frontier based optimized proposed integrated magnetic structure will be further validated by the experiments.

Simulation Validations
The parameters of the CLLC resonant converter are shown in Table 3.The simulated magnetic flux density of the conventional design, unoptimized design of the proposed integrated magnetic structure, and optimized design of the proposed integrated magnetic structure are shown in Figure 13a-c, respectively.
Based on the distributions of the magnetic flux density, the core loss of each design is calculated, which is shown in Figure 13d.It can be seen from Figure 13d that the core loss of the unoptimized design of the proposed integrated magnetic structure is higher than the conventional design.This is because the current of Lrp and Lrs flowing through the side legs of the magnetic core causes extra core loss.However, with the Pareto optimization procedure, the core loss of the optimized design of the propose integrated magnetic structure is reduced, which is lower than the conventional design.
As a result, it can be concluded that with the Pareto optimization procedure, the proposed integrated magnetic structure can enjoy higher power density and lower core loss compared with the conventional design.In the following part, the conventional design, unoptimized proposed integrated magnetic structure and the Pareto frontier based optimized proposed integrated magnetic structure will be further validated by the experiments.

Experimental Validation of the Proposed Integrated Magnetic Structure and Its Corresponding Pareto Optimization Procedure
Three planar transformers are designed for validations: conventional planar transformer structure, as shown in Figure 14a; unoptimized proposed integrated magnetic structure, as shown in Figure 14b; optimized proposed integrated magnetic structure, as shown in Figure 14c.Three planar transformers are designed for validations: conventional planar transformer structure, as shown in Figure 14a; unoptimized proposed integrated magnetic structure, as shown in Figure 14b; optimized proposed integrated magnetic structure, as shown in Figure 14c.4.
It can be seen from Table 4 that by adjusting dc and ds, all the three planar transformers share the same resonant inductances, Lrp and Lrs, and the same magnetizing inductance   4. It can be seen from Table 4 that by adjusting d c and d s , all the three planar transformers share the same resonant inductances, L rp and L rs , and the same magnetizing inductance L m .In the following part, all the three planar transformers will be tested in the CLLC resonant converter to further validate the proposed integrated magnetic structure and its corresponding Pareto optimization procedure.
The planar transformers shown in Figure 14 are tested in the CLLC resonant converter.The main switches and the rectifiers are C2M0080120D (Silicon Carbide), from CREE and the controller is TMS320F28335.The parameters of the CLLC resonant converter are the same with the parameters in Table 3.The CLLC resonant converter with the planar transformer is shown in Figure 15, where all the three transformers shown in Figure 14 will be tested.(a) Working principle validations of the proposed integrated magnetic structure In order to validate the working principles of the proposed integrated magnetic structure, the transformers in Figure 14 are tested in the CLLC resonant converter.The experimental waveforms of the output of the full bridge in the primary side, vAB(t), the output of the full bridge in the secondary side, vCD(t), and the current in the primary side, ip(t) in the three designs are shown in Figure 16, where Figure 16a shows the experimental waveforms with the conventional planar transformer; Figure 16b shows the experimental waveforms with the unoptimized proposed integrated magnetic structure; and Figure 16c shows the experimental waveforms with the optimized proposed integrated magnetic structure.
[  In order to validate the working principles of the proposed integrated magnetic structure, the transformers in Figure 14 are tested in the CLLC resonant converter.The experimental waveforms of the output of the full bridge in the primary side, v AB (t), the output of the full bridge in the secondary side, v CD (t), and the current in the primary side, i p (t) in the three designs are shown in Figure 16, where Figure 16a shows the experimental waveforms with the conventional planar transformer; Figure 16b shows the experimental waveforms with the unoptimized proposed integrated magnetic structure; and Figure 16c shows the experimental waveforms with the optimized proposed integrated magnetic structure.
experimental waveforms of the output of the full bridge in the primary side, vAB(t), the output of the full bridge in the secondary side, vCD(t), and the current in the primary side, ip(t) in the three designs are shown in Figure 16, where Figure 16a shows the experimental waveforms with the conventional planar transformer; Figure 16b shows the experimental waveforms with the unoptimized proposed integrated magnetic structure; and Figure 16c shows the experimental waveforms with the optimized proposed integrated magnetic structure.It can be seen from Figure 16a that the switching frequency of the CLLC resonant converter with the conventional planar transformer is 100 kHz, which is in accordance with the design.In addition, both the initial magnetic integration, as shown in Figure 16b, and the optimized magnetic integration, as shown in Figure 16c, share the same experimental waveforms as Figure 16a, which means the design of the resonant inductances, Lrp It can be seen from Figure 16a that the switching frequency of the CLLC resonant converter with the conventional planar transformer is 100 kHz, which is in accordance with the design.In addition, both the initial magnetic integration, as shown in Figure 16b, and the optimized magnetic integration, as shown in Figure 16c, share the same experimental waveforms as Figure 16a, which means the design of the resonant inductances, L rp and L rs , and the magnetizing inductance, L m , has been achieved by the proposed magnetic integration.However, it is important to point out that it is necessary to further investigate the power efficiency of the converter with the proposed integrated magnetic structure.
(b) The power efficiency of the CLLC resonant converter with the proposed magnetic integration It has been proven in the previous part that the proposed magnetic integration can be used for the CLLC resonant converter.In this part, the power efficiency of the CLLC resonant converter with the proposed magnetic integration will be further investigated.
The power efficiency of the CLLC resonant converter with three different transformers in Figure 14 under different output power is tested and the experimental results are shown in Figure 16d.
It can be seen from Figure 16d that compared with the conventional planar transformer, where the leakage inductances are used as the resonant inductances, the efficiency of the unoptimized proposed integrated magnetic structure is lower.This is because that the magnetic flux density increases due to the magnetic integration.As a result, both the copper loss and the core loss will increase.
In addition, it can be seen from Figure 16d that the efficiency of the CLLC resonant converter with the optimized integrated magnetic structure is higher than that with the conventional transformer.Thanks to the Pareto optimization procedure, the distributions of the magnetic flux density are optimized, and the power loss of the magnetic core is reduced.
It is worth pointing out that the selection of the design from the Pareto frontier, Point A, as shown in Figure 12, is based on the same magnetic core volume, which is used as an example.The selection of the design from the Pareto frontier is the trade-off between the volume and the power loss, which is dependent on the specific application.However, it has been proven that the proposed Pareto optimization procedure based integrated magnetic structure can achieve higher efficiency than the conventional planar transformer under the same volume.

Conclusions
An integrated magnetic structure for the CLLC resonant converter has been proposed in this paper.Two resonant inductances and a transformer are integrated into a magnetic core.Compared with the conventional planar transformer, where the leakage inductances are used as the resonant inductances, the advantages of the proposed integrated magnetic structure can be summarized as follows: Firstly, the extra height of the magnetic core, which is used to generate the desired leakage inductances in the conventional planar transformer is not required anymore.As a result, the volume of the core will be reduced.
Secondly, as the leakage flux stored between primary winding and secondary winding is reduced, the copper loss caused by the proximity effect will be improved.
Thirdly, with the proposed integrated magnetic structure, the range of the leakage inductances is extended, which can be easily changed by the number of turns in the resonant inductances.
Finally, the multi-objective optimization method, Pareto optimization procedure, is utilized for the optimal design of the proposed integrated magnetic structure and high efficiency can be achieved.
The advantages of the proposed integrated magnetic structure have been validated by the simulations and experiments.Mutual inductance between L r1 and L r2 , see (6).(H) M 1m Mutual inductance between L r1 and L m , see (6).

(H) M 2m
Mutual inductance between L r2 and L m , see (8).(H) N p , N s Number of turns in the primary winding and secondary winding, respectively, see Table 1.
A es , A ec Cross-sectional area of the side leg and central leg, respectively, see Table 1 and Figure 7. (m 2 ) k c , α, β Parameters of the magnetic core, see Tables 1 and 4. w s , w c Width of the side legs and central leg, respectively, see Figure 7. (m) W w , W h , w d Width, height and depth of the window, respectively, see Figure 7. (m) I r1pk , I r2pk , I mpk Peak values of i r1 , i r2 and i m , respectively, see (13a), (13b) and (13c).(A) B Lpk , B Rpk , B Cpk Peak values of V L , V R and V C , respectively, see (13a), (13b) and (13c).(T) P c Total core loss of the magnetic core, see (14)   v AB (t), v CD (t) Output voltage of the full bridge in the primary side and secondary side, respectively, see Figure 16.(V) i p (t) Current in the primary side, see Figure 16.(A)

Figure 2 .
Figure 2. Magnetic design for a CLLC resonant converter: (a) Detached magnetic cores; (b) conventional integrated magnetic structure; (c) proposed integrated magnetic structure.

Figure 2 .
Figure 2. Magnetic design for a CLLC resonant converter: (a) Detached magnetic cores; (b) conventional integrated magnetic structure; (c) proposed integrated magnetic structure.

Figure 2 .
Figure 2. Magnetic design for a CLLC resonant converter: (a) Detached magnetic cores; (b) conventional integrated magnetic structure; (c) proposed integrated magnetic structure.

Figure 4 .
Figure 4. Magnetic flux analysis of the proposed integrated magnetics.

Figure 4 .
Figure 4. Magnetic flux analysis of the proposed integrated magnetics.

Energies 2021 , 20 Figure 4 .
Figure 4.The flux generated by Lr2, ϕr2, is shown by the green dash line.The equivalent magnetic circuit with the excitation is shown in Figure 5.

Figure 5 .
Figure 5. Equivalent magnetic circuit with the excitation of: (a) i r1 ; (b) i r2 ; (c) i m .Derivation of L r1 , L r2 , L m , M 1m and M 2m

3. 2 .
Comparison with the Conventional Integrated Magnetic Structure for a CLLC Resonant Converter (Using the Leakage Inductances as the Resonant Inductances)

Figure 6 .
Figure 6.2D structure of the magnetics and their simulation results with: (a) Conventional structure; (b) Proposed integrated magnetic structure.

Figure 6 .
Figure 6.2D structure of the magnetics and their simulation results with: (a) Conventional structure; (b) Proposed integrated magnetic structure.

Figure 7 .
Figure 7. 3D model of a magnetic core.4.1.Derivation of the Core Loss (Pc)With ir1, ir2 and im applied to Lr1, Lr2 and Lm, the flux density of VL, VR and VC, BLpk, BRpk and BCpk, can be calculated by:

Figure 7 .
Figure 7. 3D model of a magnetic core.4.1.Derivation of the Core Loss (P c ) (a) Effect of w c on Core Loss In the first case, the width of the center leg, w c , is changed from 20 to 140% of the base value (10.4 mm), while other parameters are kept the same with the base values.Simulations are carried out in Ansys Maxwell and the distributions of the magnetic flux density are shown in Figure 8. Energies 2021, 14, x FOR PEER REVIEW 10 of 20 (a) Effect of wc on Core Loss In the first case, the width of the center leg, wc, is changed from 20 to 140% of the base value (10.4 mm), while other parameters are kept the same with the base values.Simulations are carried out in Ansys Maxwell and the distributions of the magnetic flux density are shown in Figure 8.

Figure 8 .
Figure 8. Distribution of magnetic flux density with different w c : (a) w c = 5.2 mm; (b) w c = 10.4 mm; (c) w c = 20.8mm; (d) core loss with w c .

Figure 9 .
Figure 9. Distribution of magnetic flux density with different w s : (a) w s = 52.6 mm; (b) w s =5.2 mm; (c) w s = 10.4 mm; (d) core loss with w s .

Figure 10 .
Figure 10.Distribution of magnetic flux density with different w d : (a) w d = 50.8mm; (b) w d = 101.6mm; (c) w d = 203.2mm; (d) core loss with w d .

Figure 11 . 1 : 2 :
Figure 11.Flowchart of the Pareto optimization procedure of the proposed integrated magnetic structure.The details of each step are as follows: Step 1: Initialization of Pareto optimization procedure The operation conditions of the CLLC resonant converter, O= {Ir1pk, Ir2pk, Impk}, which are derived from the circuit simulations, are set.The design space, Wd = {40.64mm ≤ wd ≤ 160.4 mm}, Wc = {4.08 mm ≤ wc ≤ 15.3 mm}, Ws= {2.08 mm ≤ ws ≤ 7.80 mm}, which are used to limit the dimensions of the magnetic core, are defined.The population, Xj = {x1, x2, …, xi, …, xN}, where xi= {wdi, wci, wsi}, are initialized.Step 2: Calculations of the dominated number of each individual (np) and crowding distance (Cd) In this step, at first, the multi-objective values, the volume of the magnetic core, Vc, and the core loss, Pc, are calculated.np and Cd of each individual are calculated with Vc and

Figure 11 .
Figure 11.Flowchart of the Pareto optimization procedure of the proposed integrated magnetic structure.(b) Application of Pareto optimization procedure for the proposed integrated magnetic structure

Figure 12 .
Figure 12.Pareto frontier of the proposed integrated magnetic structure: A: selected optimal design of the proposed integrated magnetic structure; B: conventional design; C: unoptimized design of the proposed integrated magnetic structure.

Figure 12 .
Figure 12.Pareto frontier of the proposed integrated magnetic structure: A: selected optimal design of the proposed integrated magnetic structure; B: conventional design; C: unoptimized design of the proposed integrated magnetic structure.

Figure 13 .
Figure 13.Simulation results: (a) conventional design; (b) unoptimized design of the proposed integrated magnetic structure; (c) optimized design of the proposed integrated magnetic structure; (d) power loss comparison.

Figure 13 .
Figure 13.Simulation results: (a) conventional design; (b) unoptimized design of the proposed integrated magnetic structure; (c) optimized design of the proposed integrated magnetic structure; (d) power loss comparison.

5. 2 .
Experimental Validation of the Proposed Integrated Magnetic Structure and Its Corresponding Pareto Optimization Procedure

Figure 14 .
Figure 14.The planar transformers of CLLC resonant converter for validations: (a) conventional planar transformer; (b) unoptimized proposed integrated magnetic structure; (c) optimized proposed integrated magnetic structure.Lr1, Lr2, Lrp, Lrs, and Lm are set by changing the air gaps in the central leg (dc) and the side legs (ds) and are measured by the impedance analyzer.The measured results of the inductances in each structure are shown in Table4.It can be seen from Table4that by adjusting dc and ds, all the three planar transformers share the same resonant inductances, Lrp and Lrs, and the same magnetizing inductance

Figure 14 .
Figure 14.The planar transformers of CLLC resonant converter for validations: (a) conventional planar transformer; (b) unoptimized proposed integrated magnetic structure; (c) optimized proposed integrated magnetic structure.

L
r1 , L r2 , L rp , L rs , and L m are set by changing the air gaps in the central leg (d c ) and the side legs (d s ) and are measured by the impedance analyzer.The measured results of the inductances in each structure are shown in Table

Figure 15 .
Figure 15.The CLLC resonant converter for the validation.
(a) Working principle validations of the proposed integrated magnetic structure

and 10d w co 10 .
4 mm, the original width of the central leg, see Figure 8d (m) w so 5.2 mm, the original width of the side leg, see Figure 9d.(m) w do 101.6 mm, the original depth of the window, see Figure 10d (m) V c Volume of the magnetic core, see Figure 12. (m 3 )

Table 1 .
Parameters of the CLLC resonant converter and the magnetic core.

Table 2 .
The Sizes of the magnetic core.

Table 2 .
The Sizes of the magnetic core.

Table 3 .
The Parameters of the CLLC resonant converter and the current excitations.

Table 3 .
The Parameters of The CLLC resonant converter and the current excitations.
and Figure 12. (W) f s Switching frequency, see (14) and Table 3. (Hz) V in , V o DC input and output voltage, see Table 3. (V) P o Output power, see Table 3. (W) P c / P co Ratio of P c and the original core loss P co (5.8W), see Figures 8d, 9d